2023/08/01 by Iryna Kuznietsova, Kuznietsova, Iryna, Sergiy Maksymenko +1 · 1 citation
Mathematics · #37C05 #57R45 #57S05 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2308.00577
openalex publication_date 2023/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a compact surface M, consider the natural right action of the group of diffeomorphisms D(M) of M on C∞(M,ℝ) defined by the rule: (f,h)↦ f∘ h for f∈ C∞(M,ℝ) and h\inD(M). Denote by F(M) the subset of C∞(M,ℝ) consisting of function f:M→ℝ taking constant values on connected components of ∂M, having no critical points on ∂M, and such that at each of its critical points z the function f is C∞ equivalent to some homogenenous polynomial without multiple factors. In particular, F(M) contains all Morse maps. Let also O(f) = \ f∘ h | h\inD(M) \ be the orbit of f. Previously it was computed the algebraic structure of π1O(f) for all f\inF(M), where M is any orientable compact surface distinct from 2-sphere. In the present paper we compute the group π0S(f,∂\mathbbM), where \mathbbM is a Möbius band, and S(f,∂\mathbbM) = \ h\inD(\mathbbM) | f∘ h = f, h|_∂ \mathbbM = id_\mathbbM\ is the subgroup of the corresponding stabilizer of f consisting of diffeomorphisms fixed on the boundary ∂ \mathbbM. As a consequence we obtain an explicit algebraic description of π1O(f) for all non-orientable surfaces distinct from Klein bottle and projective plane.